Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $-0.707 - 0.707i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61i·3-s i·7-s − 1.61·9-s i·11-s − 1.61·13-s − 17-s + 1.61i·19-s − 1.61·21-s + i·27-s − 29-s + 0.618i·31-s − 1.61·33-s + 2.61i·39-s + 41-s i·43-s + ⋯
L(s)  = 1  − 1.61i·3-s i·7-s − 1.61·9-s i·11-s − 1.61·13-s − 17-s + 1.61i·19-s − 1.61·21-s + i·27-s − 29-s + 0.618i·31-s − 1.61·33-s + 2.61i·39-s + 41-s i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $-0.707 - 0.707i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (2751, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4000,\ (\ :0),\ -0.707 - 0.707i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5910114934\)
\(L(\frac12)\)  \(\approx\)  \(0.5910114934\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 1.61iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - 1.61iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - 0.618iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - 0.618iT - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 + 0.618iT - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 + 0.618iT - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.618T + T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.73858769749365510443121234510, −7.60686784246973024753074085497, −6.81196306070889522316269909129, −6.16307091778900805176193829291, −5.40817734288491161454754442675, −4.32464146740364387940538477509, −3.35866264594108745200205293175, −2.34021355019288633650008374236, −1.52140698720427526983637471325, −0.30685085800694714946030010550, 2.35624710858523781555697740030, 2.65755440860477085356212857283, 3.98671781572133616016195783297, 4.68061046561209791955590972037, 5.05116724938903643651891765014, 5.87338519582092453275473412395, 6.94418992391571277579815186616, 7.60519233706503418217431995352, 8.785402740278309147073546452890, 9.189046883754691625837819802678

Graph of the $Z$-function along the critical line